‘State a rule that governs (determines/characterizes) how people do make decisions’…
I want this rule to be…
Informative (it rules out at least some sets of choices)
Predictive (people rarely if ever violate this rule)
Similar question:
‘State a rule that governs how people should make decisions’..
By ‘should’ I mean that they will not regret having made decisions in this way.
2 minutes: discuss with your neighbour
If people did follow these rules, what would this imply and predict?
Rules defined as ‘axioms about preferences’
‘Standard axioms’ \(\rightarrow\) (imply that) choices can be expressed by ‘individuals maximising utility functions subject to their budget constraints’
\(\rightarrow\) yields predictions for individual behavior, markets, etc.
Lecture 2, Ch.2 – Utility and Choice – coverage
Learn, understand, be able to explain and explain:
‘Utility’, how it’s defined
Key assumptions about preferences/choices; their implication
Depict preferences/utility with ‘indifference curves’
… examples of ‘perfect substitutes’ and ‘perfect complements’
‘Budget constraints’, compute and model them
Maximising utility subject to constraints
optimisation condition for this
Depict and interpret optimisation with indifference curves and budget constraints
Utility
Utility
“The pleasure or satisfaction that people get from their economic activity.”
Alt: The thing that people maximise when making economic decisions
How is this used?
Utility will be expressed as a single number that arises from the combination of all goods and services consumed.
Essentially, economists assume that ‘when making a choice among all available and feasible options, an individual will choose the one that yields the greatest utility’
Utility from two goods
\[Utility = U(X,Y; other)\]
Leisure and ‘goods consumption’
Food and non-food
Coffee and tea (holding all else constant)
\[Utility = U(X,Y)\]
Maths revision: \(U(X,Y)\) expresses a function with two arguments, X and Y.
\(U(X,Y)\) must take some value for every positive value of X and Y.
This expresses a general function; I haven’t specified what this function is
E.g., it could be \(U(X,Y)=\sqrt(XY)\)
Measuring and comparing utilities
Utility is not ‘observable and measurable in utils’
(Unlike midi-chlorians or thetans)
Utility is seen to govern an individual’s choices and thus it’s only inferred indirectly, from the choices people make
Revealed preference: if Al buys a cat instead of a dog, and a dog was cheaper, we assume Al gets more utility from a cat
Interpersonal comparisons are difficult
Who gets ‘more’ utility?
…Interpersonal comparisons are difficult
Transfer from Al to Betty: Is the reduction in Al’s utility more or less than the increase in Betty’s?
Standard assumptions about preferences (‘axioms’)
Completeness
Transitivity (internal consistency)
More is Better (nonsatiation)
1. Completeness
\[A \succ B, B \succ A, \: or \: A \sim B \]
Fancy notation: Either A preferred to B, B preferred to A, or A indifferent to B
Forbidden: “I can’t choose between a ski holiday and a beach holiday, but I am not indifferent”
2. Transitivity
\[ A \succ B \: and \: B \succ C \rightarrow A \succ C \]
Similar for indifference (\(\sim\))
If I prefer an Apple to a Banana and a Banana to Cherry
then I prefer an Apple to a Cherry.
If not \(\rightarrow\) money pump.
3. More is better
(similar to nonsatiation, ‘monotonicity’)
Draw this!
If the product is a ‘bad’ (e.g., pollution), redefine as the absence of the product
Who cares?
If people obey the first two assumptions (axioms),\(^\ast\) they will make choices in a way consistent with maximising a (continuous) utility function
*(and also ‘continuity’, which you can ignore for this module)
How can we compare the “?” areas? Which are preferred?
\(\rightarrow\) Compare utilities, depict using Indifference Curves
Indifference curves
Indifference curve
A curve that shows all the combinations of goods or services that provide the same level of utility
Indifference curves
Indifference curve
A curve that shows all the combinations of goods or services that provide the same level of utility
Formally (for 2 goods), the set of pairs of \(\{X,Y\}\) such that \(U(X,Y)=c\)
for some constant \(c\)
Warning: Indifference curves help us depict utility functions; a single indifference curve is not itself a utility function!
Credit: www2.econ.iastate.edu
Credit: Frank’s Economics on the web (MIT)
Properties of indifference curves
Rank order of preference/indifference between points A-E.
Q1: How do we know \(E \succ B\) ?
Q: How do we know \(E \succ A\) ?
Why ‘voluntary trade’?
The indifference curve offers some intuition.
Marginal rate of substitution (MRS)
MRS = Absolute value of slope of indifference curve
‘Rate at which you’re willing to forgo consuming \(Y\) to consume one more \(X\)’
Back to fig 2.2 (board or visualiser)
A to B: willing to give up 2 hamburgers to get 1 more soda.
\(\rightarrow\) slope \(-2\), \(MRS=2\)
From B to C? (think about it)
…willing to give up 1 hamburger to get 1 more soda \(\rightarrow MRS =1\)
C to D?
\(MRS = \frac{1}{2}\)
Note the decline: ‘diminishing MRS’: may reflect satiation
Preference for variety/balance
Indifference curve map
Indifference curves never cross! And are never upwards sloping!
Illustrating particular preferences (NS fig 2.5)
App 2.3: Product positioning in marketing (read at home, see handout)
Definitions: Perfect substitutes and complements
Perfect substitutes
Perfect substitutes
Goods A and B are Perfect Substitutes when an individual’s utility is linear in these goods
when she is always willing to trade off A for B at a fixed rate (not necessarily 1 for 1)
Perfect complements
Goods A and B are Perfect Complements when an individual only gains utility from (more) A if she also consumes a defined (additional) amount of B, and vice-versa
These goods are ‘enjoyed only in fixed proportions’.
Choices are subject to constraints :(
You cannot spend more than your (lifetime) income/wealth
\(\rightarrow\)budget constraint.
Budget constraint for two goods, slope \(-P_x/P_y\)
Budget constraint algebra
If I spend all my income (I will do over a ‘relevant lifetime’):
Expenditure on X + Expenditure on Y = Income (I)
\[P_X X + P_Y Y = I \]
To see how \(Y\) trades off against \(X\), rearrange this to:
\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]
Expenditure on X + Expenditure on Y = Income (I)
\[P_X X + P_Y Y = I \]
\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]
Intercept \(\frac{I}{P_Y}\): amount of Y you can buy if you only buy Y
Slope \(-\frac{P_X}{P_Y}\): how much Y you must give up to get another X
Utility maximization
If slope budget constraint = slope indifc curve at point X,Y \(\rightarrow\)\[P_X/P_Y = MRS(X,Y)\]
Warning: This equality holds at an optimal choice; it doesn’t hold everywhere.
At an optimal consumption choice (given above assumptions)
Consume all of income; locate on budget line; follows from ‘more is better’
Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods
Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods
Key intuition:
If I can give up X for (buy less X, get more Y) at some rate, and the benefit I get from doing this is at a different rate,
…then I can make myself better off.
Thus the original point could not have been optimal.
More insight (mathy: ignore if this freaks you out)
Recall \(U=U(X,Y)\).
\[U_X(X,Y) := MU_X(X,Y)\]
Derivative w/ respect to X: rate utility increases if we add a little X, holding Y constant
… Similarly for \(MU_Y\).
MRS: ‘how much Y would I be willing to give up to get a unit of X’?
Ans: Depends on marginal benefit of each … we can show \(MRS(X,Y)=\frac{MU_{X}}{MU_{Y}}\)
Rearranging the utility maximising condition yields more intuition:
\[P_X/P_Y = MRS = MU_X/MU_Y\]
(at each consumption point X,Y)
\[\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\]
Same ‘bang for each buck’ (if optimising)
Caveat on ‘corner solutions’
If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold
Advanced: This is a ‘necessary but not sufficient condition’, sufficient if DMRS everywhere
But…
But…
If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold
But you might consume none of some good (say X):
if even with no X, \(MU_X/P_X<MU_Y/P_Y\), the marginal utility of the first unit ‘per pound’ is lower
App 2.4: ticket scalping
App 2.5: What’s a rich uncle’s promise worth?
Using the model of choice
Why do people spend their money on different things?
What do different preferences/indifference curves imply for choices?
Move to ppt slides here beginning with ‘Utility Maximization: A Graphical View’
Algebraic/numerical examples
Consider: are these ‘perfect substitutes’ for someone who wants caffeine, but has no taste buds?
Perfect substitutes, but not identical, e.g.,
\[U(X,Y)=4X+3Y\]
Rates each increase utility per-unit (derivative) are constant: \(MU_X = 4\), \(MU_Y = 3\)
So (for perfect substitutes) buy the one that increases it more *per-
With perfect substitutes: ‘Bang for the buck’ rule
\[U(X,Y)=4X+3Y\]
Compare \(MU_X/P_X\) to \(MU_Y/P_Y\)
Here, if \(4/P_X > 3/P_Y\), then buy X
if \(4/P_X < 3/P_Y\), then buy Y; (if equal, buy either)
Rearranging, if \(P_X < 4/3 P_Y\), buy X … etc.
Warning: If not perfect substitutes, MU ratios depend on consumption levels.
Perfect complements
Mathematical function example:
\[U(X,Y)=min(2X,Y)\]
E.g., X: bicycle frames, Y: wheels.
Warning: this min function looks backwards, but it’s correct; see notes
Shortcut: figure out the proportions it will be consumed in
determine cost of ‘1 bundle of the combo’ at given prices